On this site, eclipses are named by the GMT date of their maximum eclipse; but this might not be the correct date in your timezone. Times also need to be adjusted for your timezone. Altogether dates can be more confusing than you would think, particularly when looking at historical dates. This page is here to help.
Each eclipse on this site is identified by its date in Universal Time (which, for most practical purposes, is the same as UTC, or "GMT"); this gives us a simple and consistent way to name an eclipse. However, you should be aware that the date might be different in your time zone; for example, the lunar eclipse of 28 September, 2015 actually occurs on 27 September in the "America/Los_Angeles" timezone (PDT). Also, the date we use is the date on which the moment of maximum eclipse occurs; the eclipse may actually begin on the previous day or end on the next.
To help you make sense of this, this site allows you to configure a preferred time zone for viewing eclipses. You can get to the timezone setting from the "Preferences" button in the left sidebar (where you can also see your current timezone). Once you have set a timezone, eclipse timings will be shown in that timezone as well as UT, and a warning will appear if part of a given eclipse occurs on a different day.
One word of warning — conversion to your local time is done based on this server's stored timezone rules. It's possible for those rules to be in error, particularly when governments change their daylight savings time rules at short notice (which they do). So, please take care to check that the times are correct before making serious plans.
When looking at eclipses in the past, be aware that timezones were only formally defined relatively recently; for example, the USA only passed the Standard Time Act in 1918. So even when we list a time in a given timezone, that timezone might not have actually been in use in a given place at that point in history; more likely, informal solar-based time may have been used, which should at least roughly correspond to the modern zone time. (Prior to 1582, we display times in UT only.)
For dates up to 1923, there's another wrinkle — different calendars were used in different countries. This began when a huge change was made to our calendar, from the Julian to the Gregorian calendar, in 1582. When the new Gregorian calendar was introduced, the day number suddenly jumped forwards by 10 days. So on this site, all eclipse dates up to Oct 04, 1582 are in the Julian, or Old Style calendar; and all dates from Oct 15, 1582 are shown in the Gregorian, or New Style calendar. (Wikipedia's article on the Gregorian calendar explains this in more detail.)
Unfortunately, the calendar change was implemented at different times in different places; as late as 1923 in some countries. So for 340 years, there was a transition period in which both Old Style and New Style calendars were in use in different countries. This creates potential confusion, where an event like an eclipse, which is seen over much of the Earth, may be dated according to the Julian calendar in some countries, and the Gregorian in others. To help with this, all the eclipse detail pages during the transition period list the Old Style date after the New Style date. (No attempt is made to account for the year starting on different dates; for example, on 25 March in England before 1752.)
For example, the United Kingdom and its dependencies (including the American colonies) only changed calendars in 1752. Hence, Edmond Halley wrote a detailed prediction for the eclipse in England on the 22nd of April, 1715, in the Old Style calendar which was still in use in England; that eclipse is listed here as occurring on the 3rd of May, 1715, which is the date that would have been recorded by observers in much of Europe. Wikipedia also has an article on this eclipse.
Basically, UT is UTC (also known as "GMT"); all the world's timezones are defined with respect to UTC (except where solar time is used).
More specifically... UT is the time used by astronomers, as it represents the time based on astronomical observation of the Earth's rotation, as measured at the Greenwich Meridian. Since eclipse predictions are based on the position of the Earth and Moon relative to the Sun, this is quite natural.
The problem with this is that the Earth doesn't rotate evenly; not only is it slowing down overall (don't worry, not by very much!), but the rate of rotation can actually speed up and slow down erratically, due to effects such as tidal forces. That would mean that the length of the second calculated by dividing the day into bits would vary from day-to-day.
Because we naturally want mid-day to occur when the Sun is overhead, our civil time is very close to UT time; but to keep the length of a second constant, it is also linked to atomic clock time. This is the UTC standard. So UTC ticks at exactly constant intervals, determined by the atomic clock second; but it is kept close (within 0.9 of a second) to UT by adding a leap second every year or two. So basically, for most purposes, UT is the same as UTC.
Sometimes the dates of eclipses shown here don't match those on NASA's site, particularly for eclipses in the remote past. This is because NASA gives the date of an eclipse in Dynamical Time, a.k.a. Terrestrial Time (TD, or TDT), which is a timescale based on atomic clocks, ticking at a constant rate with no leap seconds. But as described above our civil time is based on UT, which tracks the rise and set of the Sun, and so it is linked to the Earth's chaotic rotation. This means that TD and UT can differ by minutes, or even hours in the distant past or future; and so the date of a specific event can be different in the two timescales.
On this site, all eclipse dates and times are in UT, so they basically match our concept of civil time; this is why you will sometimes see a mismatch versus the Nasa dates and times. An example is the solar eclipse of 11 Oct, 1539 AD, which is listed on NASA's site as being on "1539 October 12". Looking at the "Instant of Greatest Eclipse" on that NASA page, you will see that greatest eclipse was on October 12 at 00:01:45 TDT, and that this equates to October 11 at 23:59:07 UT — the previous day.
It's fairly easy (for the experts at NASA, anyway) to predict when an eclipse will occur in atomic clock time, i.e. TD. However, to know where on Earth it will be seen, you need to know how far the Earth has rotated at the time; in other words, you need the time in UT. But as mentioned above, the Earth's rotation is erratic, and so is hard to estimate very far (centuries or more) into the past or future. The difference between the TD and UT timescales is called "ΔT", or "Delta T". An estimate of ΔT is a key part of eclipse predictions.
NASA's site lists times in TD because those are closer to being definitive; the best available estimate of ΔT can be applied, to get the UT time and location, after the basic prediction has been made up. Since this is an estimate, both the UT time and the location of an eclipse are approximate, with a potential error which increases as you go farther from the present day; for example, the time of an eclipse occurring in 1 AD could be off by 8½ minutes, and the location by over 2 degrees in longitude (120 miles or more).
You will find a box describing the uncertainty in the time and location of each eclipse in its data page. This information is derived from NASA's estimates of how big the error in ΔT could be. The good news is that for eclipses in the telescope age, i.e. the last couple of centuries, ΔT is known pretty accurately; so the UT times given here should be good to within a second or two, and the locations to within a mile or so.
As usual, note that the above information has been compiled by me from various sources, and I may well have got it wrong. However, you can always go and check up with the people who really know. Alternatively, you might just want more background information on all the systems of time in use. These links might come in handy: